Asking questions to minimize errors

Conjunctions of Unate DNF Formulas: Learning and Structure

A read-once formula is a Boolean formula in which each variable occurs, at most, once. Such formulas are also called μ-formulas or Boolean trees. This paper treats the problem of exactly identifying an unknown read-once formula using specific kinds of queries. The main results are a polynomial-time algorithm for exact identification of monotone read-once formulas using only membership queries, and a polynomial-time algorithm for exact identification of general read-once formulas using equivalence and membership queries (a protocol based on the notion of a minimally adequate teacher [1]). The results of the authors improve on Valiant's previous results for read-once formulas [26]. It is also shown, that no polynomial-time algorithm using only membership queries or only equivalence queries can exactly identify all read-once formulas.

Polynomial time learning of simple deterministic languages via queries and a representative sample

We present an algorithm to learn a deterministic timed automaton (DTA) via membership and equivalence queries. Our algorithm is an extension of the L* algorithm with a Myhill-Nerode style characterization of recognizable timed languages, which is the class of timed languages recognizable by DTAs. We first characterize the recognizable timed languages with a Nerode-style congruence. Using it, we give an algorithm with a smart teacher answering symbolic membership queries in addition to membership and equivalence queries. With a symbolic membership query, one can ask the membership of a certain set of timed words at one time. We prove that for any recognizable timed language, our learning algorithm returns a DTA recognizing it. We show how to answer a symbolic membership query with finitely many membership queries. We also show that our learning algorithm requires a polynomial number of queries with a smart teacher and an exponential number of queries with a normal teacher. We applied our algorithm to various benchmarks and confirmed its effectiveness with a normal teacher.

A formula is read-once if each variable appears at most once in it. An arithmetic read-once formula is one in which the operators are addition, subtraction, multiplication, and division. We present polynomial time algorithms for exact learning of arithmetic read-once formulas over a field. We present a membership and equivalence query algorithm that identifies arithmetic read-once formulas over an arbitrary field. We present a randomized membership query algorithm (i.e., a randomized black box interpolation algorithm) that identifies such formulas over finite fields with at least $2n + 5$ elements (where n is the number of variables) and over infinite fields. We also show the existence of nonuniform deterministic membership query algorithms for arbitrary read-once formulas over fields of characteristic 0, and division-free read-once formulas over fields that have at least $2n^{3} + 1$ elements. For our algorithms, we assume we are able to perform efficiently arithmetic operations on field elements and compute square roots in the field. It is shown that the ability to compute square roots is necessary in the sense that the problem of computing $n - 1$ square roots in a field can be reduced to the problem of identifying an arithmetic formula over n variables in that field. Our equivalence queries are of a slightly nonstandard form, in which counterexamples are required not to be inputs on which the formula evaluates to $0/0$. This assumption is shown to be necessary for fields of size $o(n/ \log n)$ in the sense that we prove there exists no polynomial time identification algorithm that uses only membership and standard equivalence queries.

A general technique is developed to obtain nonlearnability results in the model of exact learning from equivalence and membership queries. The technique is applied to show that, assuming NP not=co-NP, there does not exist a polynomial-time membership and equivalence query algorithm for exactly learning read-thrice DNF formulas-boolean formulas in disjunctive normal form where each variable appears at most three times. This result adds evidence to the conjecture that DNF is hard to learn in the membership and equivalence query model.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Angluin's L$^*$ algorithm learns the minimal deterministic finite automaton (DFA) of a regular language using membership and equivalence queries. Its probabilistic approximatively correct (PAC) version substitutes an equivalence query by numerous random membership queries to get a high level confidence to the answer. Thus it can be applied to any kind of device and may be viewed as an algorithm for synthesizing an automaton abstracting the behavior of the device based on observations. Here we are interested on how Angluin's PAC learning algorithm behaves for devices which are obtained from a DFA by introducing some noise. More precisely we study whether Angluin's algorithm reduces the noise and produces a DFA closer to the original one than the noisy device. We propose several ways to introduce the noise: (1) the noisy device inverts the classification of words w.r.t. the DFA with a small probability, (2) the noisy device modifies with a small probability the letters of the word before asking its classification w.r.t. the DFA, (3) the noisy device combines the classification of a word w.r.t. the DFA and its classification w.r.t. a counter automaton, and (4) the noisy DFA is obtained by a random process from two DFA such that the language of the first one is included in the second one. Then when a word is accepted (resp. rejected) by the first (resp. second) one, it is also accepted (resp. rejected) and in the remaining cases, it is accepted with probability 0.5. Our main experimental contributions consist in showing that: (1) Angluin's algorithm behaves well whenever the noisy device is produced by a random process, (2) but poorly with a structured noise, and, that (3) is able to eliminate pathological behaviours specified in a regular way. Theoretically, we show that randomness almost surely yields systems with non-recursively enumerable languages.

Learning conditional preference networks

Exact Learning Boolean Functions via the Monotone Theory

We introduce a new fault-tolerant model of algorithmic learning using an equivalence oracle and an incomplete membership oracle , in which the answers to a random subset of the learner's membership queries may be missing. We demonstrate that, with high probability, it is still possible to learn monotone DNF formulas in polynomial time, provided that the fraction of missing answers is bounded by some constant. Even when half the membership queries are expected to yield no additional information, our algorithm will exactly identify m -term, n -variable monotone DNF formulas with an expected O ( mn 2 ) queries. The same task has been shown to require exponential time using equivalence queries alone. Thus, this model may lead to a better understanding of the power of membership queries.

Hardness of approximate two-level logic minimization and PAC learning with membership queries

We present two algorithms that use membership and equivalence queries to exactly identify the concepts given by the union of s discretized axis-parallel boxes in d-dimensional discretized Euclidean space where there are n discrete values that each coordinate can have. The first algorithm receives at most sd counterexamples and uses time and membership queries polynomial in s and logn for any d constant. Further, all equivalence queries made can be formulated as the union of O(sdlogs) axis parallel boxes.Next, we introduce a new complexity measure that better captures the complexity of a union of boxes than simply the number of boxes and dimensions. Our new measure, s, is the number of segments in the target polyhedron where a segment is a maximum portion of one of the sides of the polyhedron that lies entirely inside or entirely outside each of the other halfspaces defining the polyhedron. We then present an improvement of our first algorithm that uses time and queries polynomial in s and logn. The hypothesis class used here is decision trees of height at most 2sd. Further we can show that the time and queries used by this algorithm are polynomial in d and logn for s any constant thus generating the exact learnability of DNF formulas with a constant number of terms. In fact, this single algorithm is efficient for either s or d constant.

We investigate the learning problem of two-tape deterministic finite automata (2-tape DFAs) from queries and counterexamples. Instead of accepting a subset of ∑*, a 2-tape DFA over an alphabet ∑ accepts a subset of ∑* × ∑*, and therefore, it can specify a binary relation on ∑*. In [3] Angluin showed that the class of deterministic finite automata (DFAs) is learnable in polynomial time from membership queries and equivalence queries, namely, from a minimally adequate teacher (MAT).

We study the learning models defined by Angluin et al. (1997): learning with equivalence and limited membership queries and learning with equivalence and malicious membership queries. We show that if a class of concepts that is closed under projection is learnable in polynomial time using equivalence and (standard) membership queries then it is learnable in polynomial time in the above models. This closes the open problems by Angluin et al. (1997). Our algorithm can also handle errors in the equivalence queries.

We investigate regular tree languages exact learning from positive examples and membership queries. Input data are trees of the language to infer. The learner computes new trees from the inputs and asks to the oracle whether or not they belong to the language. From the answers, the learner may ask further membership queries until he finds the correct grammar that generates the target language. This paradigm was introduced by Angluin in the seminal work [1] for the case of regular word language. Neither negative examples, equivalence queries nor counter examples are allowed in this paradigm. We describe an efficient algorithm which is polynomial in the size of the examples for learning the whole class of regular tree languages. The convergence is insured when the set of examples contains a representative sample of the language to guess. A finite subset e of a regular tree language £ is representative for L if every transition of the minimal tree automaton for L is used at least once for the derivation of an element of the set e.